Internal monotone-light factorization for categories via preorders

Joao Xarez

It is shown that, for a finitely-complete category C
with coequalizers of kernel pairs, if every product-regular epi is
also stably-regular then there exist the reflections
(R)Grphs(C) --> (R)Rel(C),
from (reflexive) graphs into (reflexive) relations in
C, and Cat(C) --> Preord(C),
from categories into preorders in C. Furthermore, such
a sufficient condition ensures as well that these reflections do
have stable units. This last property is equivalent to the
existence of a monotone-light factorization system, provided there
are sufficiently many effective descent morphisms with
domain in the respective full subcategory. In this way, we have
internalized the monotone-light factorization for small categories
via preordered sets, associated with the reflection
Cat --> Preord, which is now just the
special case C = Set.